Question

The number of divisors of 360 (excluding 1 and 360) is:

• A. 22
• B. 24
• C. 20
• D. 18

Hint:

Always check if the question asks for "proper divisors" or excludes specific ones like 1 and the number itself.

Answer 1

The Correct Option is A

Step 1: Understand the Concept
To find the total number of divisors (factors) of any composite number $N$, we first express the number as a product of its prime factors (prime factorization). If the prime factorization is $N = p_1^{a} \times p_2^{b} \times p_3^{c} \dots$, then the total number of divisors is calculated using the formula:
Total Divisors = $(a + 1)(b + 1)(c + 1)\dots$
This formula accounts for all possible combinations of the prime factors, including the number 1 and the number itself.

Step 2: Analysis of the Number 360
First, we perform the prime factorization of 360:
$360 = 2 \times 180 = 2^2 \times 90 = 2^3 \times 45 = 2^3 \times 3^2 \times 5^1$.
Here, the exponents are $a=3$, $b=2$, and $c=1$.
Applying the formula to find the total divisors:
Total Divisors = $(3 + 1)(2 + 1)(1 + 1)$
Total Divisors = $4 \times 3 \times 2 = 24$.

Step 3: Conclusion & Final Adjustment
The total count of 24 divisors includes every factor from 1 to 360. However, the question specifically asks for the number of divisors excluding 1 and 360. Therefore, we must subtract these 2 specific divisors from our total count:
Required number of divisors = $24 - 2 = 22$.
Thus, there are 22 divisors of 360 when excluding the number 1 and the number itself.

Final Answer: (A)